Fundamentals of Convex Analysis

• Introduction: Notation, Elementary Results.- Convex Sets: Generalities; Convex Sets Attached to a Convex Set; Projection onto Closed Convex Sets; Separation and Applications; Conical Approximations of Convex Sets.- Convex Functions: Basic Definitions and Examples; Functional Operations Preserving Convexity; Local and Global Behaviour of a Convex Function; First- and Second-Order Differentiation.- Sublinearity and Support Functions: Sublinear Functions; The Support Function of a Nonempty Set; Correspondence Between Convex Sets and Sublinear Functions.- Subdifferentials of Finite Convex Functions: The Subdifferential: Definitions and Interpretations; Local Properties of the Subdifferential; First Examples; Calculus Rules with Subdifferentials; Further Examples; The Subdifferential as a Multifunction.- Conjugacy in Convex Analysis: The Convex Conjugate of a Function; Calculus Rules on the Conjugacy Operation; Various Examples; Differentiability of a Conjugate Function.

From the reviews of the first edition: "...This book is an abridged version of the book "Convex Analysis and Minimization Algorithms" (shortly CAMA) written in two volumes by the same authors... . The authors have extracted from CAMA Chapters III-VI and X, containing the fundamentals of convex analysis, deleting material seemed too advanced for an introduction, or too closely attached to numerical algorithms. Each Chapter is presented as a "lesson" treating a given subject in its entirety, completed by numerous examples and figures. So, this new version becomes a good book for learning and teaching of convex analysis in finite dimensions..." S. Mititelu in "Zentralblatt fur Mathematik und ihre Grenzgebiete", 2002 "I believe that the book under review will become the standard text doing much to implement the type of course Victor Klee was advocating and covering as it does the considerable recent development of the subject. ... If you are looking for a well-designed text for a course on convex analysis, preliminary to one on optimization or nonlinear analysis then this is the one which will certainly be a standard for many years." (John Giles, The Australian Mathematical Society Gazette, Vol. 29 (2), 2002)